Polynomials Related with Hermite-Pade Approximations to the Exponential Function.211 Modelling, Analysis and Simulation

摘要

The authors investigate the polynomials P(sub n), Q(sub m) and R(sub s), having211u001edegrees n, m and s respectively, with P(sub n) monic, that solve the 211u001eapproximation problem. The authors give a connection between the coefficients of 211u001eeach of the polynomials P(sub n), Q(sub m), and R(sub s) and certain 211u001ehypergeometric functions, which leads to a simple expression for Q(sub m), in the 211u001ecase n = 2. The approximate location of the zeros of Q(sub m), when n greater 211u001ethan m and n = s, are deduced from the zeros of the classical Hermite polynomial. 211u001eContour integral representations of P(sub n), Q(sub m), R(sub s) and E(sub nms) 211u001eare given and, using saddle point methods, the authors derive the exact 211u001easymptotics of P(sub n), Q(sub m) and R(sub s) as n, m and s end to infinity 211u001ethrough certain ray sequences. The authors also discuss aspects of the more 211u001ecomplicated uniform asymptotic methods for obtaining insight into the zero 211u001edistribution of the polynomials, and the authors give an example showing the 211u001ezeros of the polynomials P(sub n), Q(sub m), and R(sub s) for the case n = s = 211u001e40, m = 45.